3.6.16 \(\int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) [516]

3.6.16.1 Optimal result

Integrand size = 21, antiderivative size = 116 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \]

output

-2*a^(3/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d+(a-I*b)^(3/2)*arctanh 
((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d+(a+I*b)^(3/2)*arctanh((a+b*tan(d* 
x+c))^(1/2)/(a+I*b)^(1/2))/d
 

3.6.16.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {-2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+(a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+(a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \]

input

Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2),x]
 

output

(-2*a^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + (a - I*b)^(3/2)*Ar 
cTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (a + I*b)^(3/2)*ArcTanh[Sq 
rt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d
 

3.6.16.3 Rubi [A] (warning: unable to verify)

Time = 0.95 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4056, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan (c+d x)}dx\)

\(\Big \downarrow \) 4056

\(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+a^2 \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx\)

\(\Big \downarrow \) 4022

\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} i (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} i (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} i (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} i (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\)

\(\Big \downarrow \) 4020

\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {(a-i b)^2 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {(a+i b)^2 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\)

\(\Big \downarrow \) 25

\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+\frac {(a-i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {(a+i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\)

\(\Big \downarrow \) 73

\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {i (a+i b)^2 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {i (a-i b)^2 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\)

\(\Big \downarrow \) 221

\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+\frac {i (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {a^2 \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}+\frac {i (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 a^2 \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {i (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {i (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\)

input

Int[Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2),x]
 

output

(I*(a - I*b)^(3/2)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/d - (I*(a + I*b)^(3 
/2)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/d - (2*a^(3/2)*ArcTanh[Sqrt[a + b* 
Tan[c + d*x]]/Sqrt[a]])/d
 

3.6.16.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4020

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f)   Subst[Int[(a + (b/d)*x)^m/(d^2 + 
c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ 
b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
 

rule 4022

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2   Int[(a + b*Tan[e + f*x])^m*( 
1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2   Int[(a + b*Tan[e + f*x])^m 
*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c 
 - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]
 

rule 4056

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/((c_.) + (d_.)*tan[(e_.) 
+ (f_.)*(x_)]), x_Symbol] :> Simp[1/(c^2 + d^2)   Int[Simp[a^2*c - b^2*c + 
2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e + f*x 
]], x], x] + Simp[(b*c - a*d)^2/(c^2 + d^2)   Int[(1 + Tan[e + f*x]^2)/(Sqr 
t[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
 

rule 4117

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
 Simp[A/f   Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; 
FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
 

3.6.16.4 Maple [F(-1)]

Timed out.

hanged

input

int(cot(d*x+c)*(a+b*tan(d*x+c))^(3/2),x)
 

output

int(cot(d*x+c)*(a+b*tan(d*x+c))^(3/2),x)
 

3.6.16.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 758 vs. \(2 (90) = 180\).

Time = 0.31 (sec) , antiderivative size = 1531, normalized size of antiderivative = 13.20 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]

input

integrate(cot(d*x+c)*(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")
 

output

[1/2*(d*sqrt((a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) 
)/d^2)*log(-(3*a^4 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) + (d^3*sqrt 
(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^3 - a*b^2)*d)*sqrt((a^3 - 3*a* 
b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) - d*sqrt((a^3 - 
3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)*log(-(3*a^4 + 
 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) - (d^3*sqrt(-(9*a^4*b^2 - 6*a^2 
*b^4 + b^6)/d^4) - (3*a^3 - a*b^2)*d)*sqrt((a^3 - 3*a*b^2 + d^2*sqrt(-(9*a 
^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) - d*sqrt((a^3 - 3*a*b^2 - d^2*sqrt(- 
(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)*log(-(3*a^4 + 2*a^2*b^2 - b^4)*sq 
rt(b*tan(d*x + c) + a) + (d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + ( 
3*a^3 - a*b^2)*d)*sqrt((a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + 
 b^6)/d^4))/d^2)) + d*sqrt((a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b 
^4 + b^6)/d^4))/d^2)*log(-(3*a^4 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + 
a) - (d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^3 - a*b^2)*d)*sq 
rt((a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) + 
2*a^(3/2)*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/ 
tan(d*x + c)))/d, 1/2*(4*sqrt(-a)*a*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(- 
a)/a) + d*sqrt((a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^ 
4))/d^2)*log(-(3*a^4 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) + (d^3*sq 
rt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^3 - a*b^2)*d)*sqrt((a^3 -...
 

3.6.16.6 Sympy [F]

\[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )}\, dx \]

input

integrate(cot(d*x+c)*(a+b*tan(d*x+c))**(3/2),x)
 

output

Integral((a + b*tan(c + d*x))**(3/2)*cot(c + d*x), x)
 

3.6.16.7 Maxima [F]

\[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right ) \,d x } \]

input

integrate(cot(d*x+c)*(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")
 

output

integrate((b*tan(d*x + c) + a)^(3/2)*cot(d*x + c), x)
 

3.6.16.8 Giac [F(-1)]

Timed out. \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]

input

integrate(cot(d*x+c)*(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")
 

output

Timed out
 

3.6.16.9 Mupad [B] (verification not implemented)

Time = 4.98 (sec) , antiderivative size = 2260, normalized size of antiderivative = 19.48 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]

input

int(cot(c + d*x)*(a + b*tan(c + d*x))^(3/2),x)
 

output

- atan((32*a*b^15*(a + b*tan(c + d*x))^(1/2)*(a^3/(4*d^2) - (b^3*1i)/(4*d^ 
2) - (3*a*b^2)/(4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2))/((a*b^17*16i)/d + (64* 
a^2*b^16)/d - (a^3*b^15*128i)/d - (128*a^4*b^14)/d + (a^5*b^13*96i)/d - (1 
92*a^6*b^12)/d + (a^7*b^11*384i)/d + (a^9*b^9*144i)/d) - (a^2*b^14*(a + b* 
tan(c + d*x))^(1/2)*(a^3/(4*d^2) - (b^3*1i)/(4*d^2) - (3*a*b^2)/(4*d^2) + 
(a^2*b*3i)/(4*d^2))^(1/2)*64i)/((a*b^17*16i)/d + (64*a^2*b^16)/d - (a^3*b^ 
15*128i)/d - (128*a^4*b^14)/d + (a^5*b^13*96i)/d - (192*a^6*b^12)/d + (a^7 
*b^11*384i)/d + (a^9*b^9*144i)/d) - (96*a^3*b^13*(a + b*tan(c + d*x))^(1/2 
)*(a^3/(4*d^2) - (b^3*1i)/(4*d^2) - (3*a*b^2)/(4*d^2) + (a^2*b*3i)/(4*d^2) 
)^(1/2))/((a*b^17*16i)/d + (64*a^2*b^16)/d - (a^3*b^15*128i)/d - (128*a^4* 
b^14)/d + (a^5*b^13*96i)/d - (192*a^6*b^12)/d + (a^7*b^11*384i)/d + (a^9*b 
^9*144i)/d) + (96*a^5*b^11*(a + b*tan(c + d*x))^(1/2)*(a^3/(4*d^2) - (b^3* 
1i)/(4*d^2) - (3*a*b^2)/(4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2))/((a*b^17*16i) 
/d + (64*a^2*b^16)/d - (a^3*b^15*128i)/d - (128*a^4*b^14)/d + (a^5*b^13*96 
i)/d - (192*a^6*b^12)/d + (a^7*b^11*384i)/d + (a^9*b^9*144i)/d) + (a^6*b^1 
0*(a + b*tan(c + d*x))^(1/2)*(a^3/(4*d^2) - (b^3*1i)/(4*d^2) - (3*a*b^2)/( 
4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2)*576i)/((a*b^17*16i)/d + (64*a^2*b^16)/d 
 - (a^3*b^15*128i)/d - (128*a^4*b^14)/d + (a^5*b^13*96i)/d - (192*a^6*b^12 
)/d + (a^7*b^11*384i)/d + (a^9*b^9*144i)/d) - (288*a^7*b^9*(a + b*tan(c + 
d*x))^(1/2)*(a^3/(4*d^2) - (b^3*1i)/(4*d^2) - (3*a*b^2)/(4*d^2) + (a^2*...